We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by √(3) , with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of S^2× S^1(λ ) arising from these spaces have length parameter λ≤ (√(3))^-1 , which supports a recent conjecture of Chang-Yang-Zhang[18]. Finally, static horizons with hyperbolic infinity and non-negative Chruściel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia [29], we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang [15]. Using optimal coefficients for the sub-static Heintze-Karcher inequality from [24], we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi-Zhu [52].
We study the conditional measures of every invariant probability measure for every C^1+γ diffeomorphism of a closed Riemannian manifold, and we show that they always admit an asymptotic local product structure. This is a direct application of the notion of tubular dimension which we introduce and study in the manuscript. As an additional application we prove a bound on any two consecutive conditional entropies in terms of volume growth.
For a symplectic manifold M with the Hamiltonian action of a compact Lie group G, we give a new perspective on the problem of unitarity in the quantization commutes with reduction correspondence. We argue that this correspondence leads naturally to the introduction of what we call Fourier polarizations, whose real directions include the Hamiltonian vector fields of all invariant functions of the moment map. For a Fourier polarization, quantization commutes with reduction unitarily. For a general G-invariant polarization, 𝒫 , unitarity in the quantization commutes with reduction correspondence can then be formulated in terms of the existence of a Fourier polarization, 𝒫_F , such that the quantizations with respect to 𝒫 and to 𝒫_F are unitarily equivalent. We describe this perspective in detail for symplectic toric manifolds, by considering symplectic reductions associated to the action of a subtorus T^p⊂ T^n . We consider half-form quantization in Fourier (mixed) polarizations 𝒫_∞ , whose real directions are generated by (the Hamiltonian vector fields of the) components of the T^p moment map. We show that Fourier mixed polarizations of this type can be obtained by Mabuchi geodesic rays of toric Kähler structures, starting at a given initial structure and with velocity given by the norm square of the moment map of the torus subgroup. We lift these geodesic rays to the quantum bundle via generalized coherent state transforms, which define equivariant isomorphisms between quantum Hilbert spaces for the Kähler and Fourier polarizations, where the later are obtained at infinite geodesic time. Unitarity in quantization commutes with reduction in these cases then becomes equivalent to the unitarity of these isomorphisms. In general, however, in the toric case, these isomorphisms are not unitary.
Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate $$\mathbb {Z}^d$$ Z d -map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site $${\textbf{p}}^*\in \mathbb {Z}^d$$ p ∗ ∈ Z d and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site $${\textbf{p}}^*$$ p ∗ converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site $${\textbf{p}}^*$$ p ∗ .
We study the phase transition and critical phenomenon for the grand canonical Phi(3) measure in two-dimensional Euclidean quantum field theory. The study of this measure was initiated by Jaffe, Bourgain, and Carlen-Frohlich-Lebowitz, primarily in regimes far from criticality. We identify a critical chemical potential and show that the measure exhibits a phase transition at this critical threshold. At the critical threshold, the analysis is based on establishing the correlation decay of the Gaussian fluctuations in the partition function, combined with a coarse-graining argument to show divergence of the maximum of an approximating Gaussian process.
We study the phase transition and critical phenomenon for the grand canonical Φ ^3 measure in two-dimensional Euclidean quantum field theory. The study of this measure was initiated by Jaffe, Bourgain, and Carlen–Fröhlich–Lebowitz, primarily in regimes far from criticality. We identify a critical chemical potential and show that the measure exhibits a phase transition at this critical threshold. At the critical threshold, the analysis is based on establishing the correlation decay of the Gaussian fluctuations in the partition function, combined with a coarse-graining argument to show divergence of the maximum of an approximating Gaussian process.
Consider two agents, Alice and Bob, each of whom takes a quantum input, operates on a shared quantum system K, and produces a quantum output. Alice and Bob’s operations may commute, in the sense that the joint input-output behaviour is independent of the order in which they access K. Here we ask whether this commutation property implies that K can be split into two factors on which Alice and Bob act separately. The question can be regarded as a “fully quantum” generalisation of a problem posed by Tsirelson, who considered the case where Alice and Bob’s inputs and outputs are classical. In this case, the answer is negative in general, but it is known that a factorisation exists in finite dimensions. Here we show the same holds in the fully quantum case, i.e., commuting operations factorise, provided that all input systems are finite-dimensional.
We study the relations between Blot–Shadrin–Singh tautological relation and the DR formula for Hodge character class ch _2g-1(𝔼) on ℳ_g . In particular, we prove the equivalence between Blot–Shadrin–Singh tautological relation on ℳ_g,1 and the DR formula for Hodge character class ch _2g-1(𝔼) on ℳ_g . As two applications, we first find a new push-forward tautological relation on ℳ_g+1 for g≥ 1 , which extends the Liu-Pandharipande relation to a lower degree. Second, we obtain a new partial differential equation for higher genus descendant Gromov–Witten invariants of any smooth projective variety.
In our previous work (Kiriki, S., Li, X., Nakano, Y., Soma, T in Comm. Math. Phys. 1241–1269 (2022)), we proved the existence of observable Lyapunov irregular sets for a class of surface diffeomorphisms. This correction addresses a gap in the proof of Lemma 4.5 of that paper, which arose from an oversight in applying an estimate. We rectify this by adjusting the initial setup, including the conditions on the diffeomorphism and the definitions of related sequences. These modifications secure the proof of Lemma 4.5 and, consequently, reinforce the main results of (Kiriki, S., Li, X., Nakano, Y., Soma, T in Comm. Math. Phys. 1241–1269 (2022)).
Pirogov–Sinai theory is a well-developed method for understanding the low-temperature phase diagram of statistical mechanics models on lattices. Motivated by physical and algorithmic questions beyond the setting of lattices, we develop a combinatorially flexible version of Pirogov–Sinai theory for the hard-core model of independent sets on bipartite graphs. Our results illustrate that the main conclusions of Pirogov–Sinai theory can be obtained in significantly greater generality than that of ℤ^d . The main ingredients in our generalization are combinatorial and involve developing appropriate definitions of contours based on the notion of cycle basis connectivity. This is inspired by works of Timár and Georgakopoulos–Panagiotis.
This paper is to establish a natural connection of quantum affine algebras with quantum vertex algebras. Among the main results, we construct a family of & hstrok;-adic quantum vertex algebras VL[[& hstrok;]]eta as deformations of the lattice vertex algebras VL, and establish a natural connection between twisted (and untwisted) quantum affine algebras of type A, D, or E and equivariant phi-coordinated quasi modules for VL[[& hstrok;]]eta with certain specialized eta.
In a fractional Sobolev space Hs(R2) with s <= 74, we prove the low-regularity ill-posedness for the 2D compressible Euler equations and the 2D ideal compressible MHD system. Our ill-posedness results match the H74 regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.
We give a generalization of the Atiyah-Schmid dimension formula for projective tempered representations. Then we prove the Atiyah-Schmid dimension formula for arithmetic subgroups of real reductive groups.
For quasi-periodic Schrödinger operators with small Gasymov-type potentials and arbitrary irrational frequencies, we establish a complete spectral characterization: the spectrum coincides with that of the discrete free Laplacian. Our result is non-perturbative in the sense that the smallness condition is independent of the frequency. Furthermore, we prove the absence of both residual and point spectra, thereby establishing purely continuous spectrum. The proof combines Avila’s global theory, non-perturbative almost reducibility, and Green’s function estimates.
We consider the Gibbs measure for the focusing nonlinear Schrödinger equation on the one-dimensional torus 𝕋 , that was introduced in a seminal paper by Lebowitz et al. (J Stat Phys 50(3):657—687, 1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz et al. (J Stat Phys 50(3):657—687, 1988). Its existence is however striking in view of a series of negative results by McKean (Commun Math Phys 168(3):479—491, 1995) and Rider (Commun Pure Appl Math 55(10):1231—1248, 2002).
We extend methods of Ding and Smart (Invent Math 219(2):467–506, 2020) which showed Anderson localization for certain random Schrödinger operators on ℓ ^2(ℤ^2) via a quantitative unique continuation principle and Wegner estimate. We replace the requirement of identical distribution with the requirement of a uniform bound on the essential range of potential and a uniform positive lower bound on the variance of the variables giving the potential. Under those assumptions, we recover the unique continuation and Wegner lemma results, using Bernoulli decompositions and modifications of the arguments therein. This leads to a localization result at the bottom of the spectrum.
